1. Field of the Invention
The present invention relates to a method of blind equalization in a receiver and particularly to an adaptive multi-modulus equalization method.
2. Description of the Related Art
Signals transmitted through a real voice-band channel suffer from non-ideal channel characteristics such as Additive White Gaussian Noise (AWGN), Inter Symbol Interference (ISI), fading, and phase distortion. These non-ideal channel characteristics seriously degrade and distort the original signals. When a signal passes through a particular channel, the receiver can model the effect of the channel as a filter with a transfer function H(z). To overcome the non-idea channel characteristics, the receiver thus designs an adaptive filter with a transfer function H−1  (z). H−1 (z) is the inverse transfer function of H(z), and multiplying H−1 (z) to the signal with channel effect recovers the original signal by canceling H(z). The expected response of combining the adaptive filter with the real voice-band channel is an impulse response in the time domain, and constant over all frequency range in the frequency domain. The adaptive filter described is referred to as an equalizer. An ideal equalizer recovers signals passed through the real voice-band channel, and completely removes the channel effect.
The coefficients of the equalizer, also referred as tap weights, determine the transfer function of the equalizer. The tap weights need to be adjusted and updated frequently to minimize error at the output of the equalizer. This error is effectively a measure of the difference between the actual output of the equalizer and the expected output. Generally speaking, there are two ways of acquiring new tap weights for the equalizer. One is to transmit a training sequence known by both transmitter and receiver at the beginning of the communication. The receiver then detects the impulse response of the channel from the training sequence, and obtains the tap weights by computing the inverse transfer function of the channel. The other way is to predetermine an initial value for each of the tap weights, and design a cost function according to the characteristics of the received signal. The tap weights are continually adjusted by reducing the cost of the cost function until the error is minimized (i.e. until the equalizer converges). Equalizers implementing the second technique described above are referred to as “blind equalizers”.
The channel effect varies, and an adaptive equalizer with adjustable transfer function is required to adapt any instantaneous change in the channel effect. The characteristics of the channel change slowly with temperature, movement of the receiver, and many other environmental factors. The adaptive equalizer continuously updates its transfer function by adjusting the tap weights to compensate for current channel effect. Blind equalization is considered more effective than non-blind equalization due to its ability to update the tap weights at any time without waiting for a training sequence.
Constant modulus algorithm (CMA) is a well-known technique used in blind equalization. The CMA algorithm converges an equalized signal on a constellation diagram with constellation points scattered evenly over several concentric circles. An attribute of the CMA algorithm is that the blind equalizer does require the number of valid coordinates on the constellation diagram.
The CMA algorithm defines a cost function to estimate channel noise in a received signal. The higher the output (cost) of the cost function, the larger the channel noise in the received signal. The equalizer first calculates an equalized signal by adding the products of the received signal and the tap weights. After obtaining the equalized signal, the cost function calculates the cost of the equalized signal. The cost indicates the noise level of the received signal, and this cost is used to adjust the tap weights of the equalizer. The equalizer then calculates a new equalized signal using the updated tap weights, and obtains a new cost from the new equalized signal. The cost of the cost function is expected to be reduced by repeating the above processes. The lower the cost, the lower the noise in the received signal.
The cost function of pth order (p is an integer greater than zero) is given by:D(p)=E[(|zn|p−Rp)2]  Equ. (1)where Zn is the output (equalized signal) of the equalizer, and Rp is a positive constant. The equation used to calculate the least-mean-squares (LMS) error can also be the cost function for adjusting the tap weights. The LMS error is given by:Error=E[(zn−an)2]  Equ. (2)
While there is a great similarity between equation (1) and equation (2), an expected output an of the equalizer, the original signal without channel noise, must be known in advance when using the LMS cost function. The LMS cost function is therefore not suitable for a monotonic demodulating system, as the original signal is unknown to the receiver.
An advantage of using the pth order cost function is the ability to achieve convergence without knowing the original signal in advance. By using the pth order cost function, Rp is assumed to be the ideal output of the equalizer. However, pth order cost function minimizes the difference between Rp and |Zn|P, which takes longer to achieve convergence than the LMS technique. Another drawback to Pth order cost function is that carrier phase distortion cannot be recovered.
FIG. 1 is a diagram illustrating the ISI effect on a signal. ISI is considered to cause the most serious distortion to the signal compared to other types of channel noise. Locations of white dots therein represent coordinates of an original signal without ISI interference on a constellation diagram. The original signal interferes with ISI when transmitting on a physical channel, causing the coordinates of the received signal to shift locations of black dots on the constellation diagram. As shown in the diagram, the black dots represent coordinates of the signal with ISI interference. The cost, that is the output of the cost function, of the original signal is 2a2, whereas the cost of the signal with ISI is 2a2+4c2. The difference of 4c2 indicates that the cost of the signal with ISI is greater than that of the original signal.
Adjusting the coefficients (tap weights) of the equalizer can reduce costs and cancel the effects of ISI interference. The steepest gradient descent method is a possible solution for adjusting the tap weights to minimize the cost of the cost function. Equations (3), (4), and (5) show the formula for adjusting the tap weights.                               c                      n            +            1                          =                              c            n                    -                                                    λ                p                            ⁡                              [                                                      ∂                                          D                                              (                        p                        )                                                                                                  ∂                    c                                                  ]                                                    c              =                              c                n                                                                        Equ        .                                   ⁢                  (          3          )                                                                                                                                  p                    =                    1                                                                                                                                                                                                       c                                              n                        +                        1                                                              =                                                                  c                        n                                            -                                                                        λ                          1                                                ⁢                                                                              y                            n                                                    ·                                                                                    z                              n                                                        ⁡                                                          (                                                              1                                -                                                                                                      R                                    1                                                                                                                                                                                  z                                      n                                                                                                                                                                                                      )                                                                                                                                                                                                                                                                  wherein                ⁢                                                                   ⁢                                  R                  1                                            =                                                E                  ⁡                                      (                                          a                      m                      2                                        )                                                                    E                  ⁡                                      (                                                                                        a                        m                                                                                    )                                                                                                          Equ        .                                   ⁢                  (          4          )                                                                                                                                  p                    =                    2                                                                                                                                                                                                       c                                              n                        +                        1                                                              =                                                                  c                        n                                            -                                                                        λ                          2                                                ⁢                                                                              y                            n                                                    ·                                                                                    z                              n                                                        ⁡                                                          (                                                                                                                                                                                                              z                                      n                                                                                                                                            2                                                                -                                                                  R                                  2                                                                                            )                                                                                                                                                                                                                                                                  wherein                ⁢                                                                   ⁢                                  R                  2                                            =                                                E                  ⁡                                      (                                                                                                                    a                          m                                                                                            4                                        )                                                                    E                  ⁡                                      (                                                                                                                    a                          m                                                                                            2                                        )                                                                                                          Equ        .                                   ⁢                  (          5          )                    
where cn is a vector of the tap weights, λ1 and λ2 are adjusting coefficients, yn is an input signal, zn is an equalized signal (output of the equalizer), Rp is a positive number determined by the pattern of the constellation diagram, and am represents the exact coordinates on the constellation diagram. Simulations show that the convergence of tap weights is faster and more accurate when using the second order (P=2) cost function rather than the first order (p=1) cost function.
Drawbacks of the CMA algorithm include the performance of convergence for a Quadrature Amplitude Modulation (QAM) signal degrading with the number of valid coordinates on the constellation diagram. FIG. 2 shows the output of a CMA equalizer for a signal modulated by 896-QAM. There are 896 valid coordinates on the constellation diagram for a 896-QAM signal. As shown in FIG. 2, detecting each constellation point on the constellation diagram becomes impossible if the number of valid coordinates is too large.